When changing an improper integral with a negative infinity bottom bound and a positive infinity top bound to a proper integral with limit statements, the common process is to pick an x-value that is easy to work with and separate the integral into two separate improper integrals. After this step you create two separate limit statements to "turn them into proper integrals." The evaluation is easy after this.
My question is this: instead of separating the original improper integral (with two infinite bounds, one negative and one positive) into two integrals, is it acceptable to create a limit statement that makes one of the bounds proper, and then take another limit statement of that first limit statement and the integral to make the second bound proper? Or in other words, instead of making two integrals is it acceptable to do this:
Lim (Lim (Integral(f(x)dx,a,b)))
a->-∞ b->∞
My question is this: instead of separating the original improper integral (with two infinite bounds, one negative and one positive) into two integrals, is it acceptable to create a limit statement that makes one of the bounds proper, and then take another limit statement of that first limit statement and the integral to make the second bound proper? Or in other words, instead of making two integrals is it acceptable to do this:
Lim (Lim (Integral(f(x)dx,a,b)))
a->-∞ b->∞
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YES as long as you do the limit process one at a time...
for if one does not exist then the integral diverges ;
eg : if F ' ( t ) = f(t) then int over ( - ∞ , w]U[w , ∞ ) of { f(t) dt } = F (w) - lim F(a) - F(w) + lim F(b)
and if either lim does not exist then divergence occurs
for if one does not exist then the integral diverges ;
eg : if F ' ( t ) = f(t) then int over ( - ∞ , w]U[w , ∞ ) of { f(t) dt } = F (w) - lim F(a) - F(w) + lim F(b)
and if either lim does not exist then divergence occurs